Superstability of the d'Alembert functional equation in L_p^+ spaces - Przybyła, Maciej J. (Annales Mathematicae Silesianae, 2004)
Let (X,+,-,0,Σ,μ) be an abelian complete measurable group with μ(X)>0. Let f: X→ℂ be a function. We will show that if A(f)∈Lp+(X×X,ℂ) where A{f)(x,y) = f{x+y) + f(x-y) - 2f(x)f(y), x,y∈X, then f∈Lp+(X,ℂ) or there exists exactly one function g: X→ℂ with g{x+y) + g(x-y) - 2g(x)g(y), x,y∈X such that f is equal to g almost everywhere with respect to the measure μ. Lp+ denotes the space of all functions for which the upper integral of ∥f∥p is finite.